I am a physics student, and I am reading a paper in topological condensed matter employing the classifying space of Clifford Algebra.This paper
In particular, I feel that I am a bit confused by the notation used around Table 7 and equation (38). Which is about the difference between the direct product and direct sum of Clifford Algebra.
Below is my understanding, when we have two real Clifford Algebra $Cl_{0,q}$ and $Cl_{0,m}$, whose generators commute with each other, their relation is denoted by direct product, as used in equation (38). $$Cl_{0,q}\otimes Cl_{0,m}$$ Now I want to extent this algebra $$Cl_{0,q}\otimes Cl_{0,m}\rightarrow Cl_{0,q+1}\otimes Cl_{0,m}$$
The possible extension is given by the classifying space of this algebra. Here is my question
- What is the classifying space of the direct product of algebra?
- In Table 7, the author seems to use direct sum of algebra, $$ Cl_{0,q}\oplus Cl_{0,q}$$ And he claims that the to extent this guy to $Cl_{0,q+1}\oplus Cl_{0,q+1}$, the classifying space is given by $$R_{q}\times R_q$$
I don't understand what the $\times$ between two $R_q$ means. And I don't understand why the author use direct sum of two Clifford Algebra, what is the difference between the direct sum and direct product of two algebra?